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Average-Cost Pricing, Increasing Returns, and Optimal Output in a Model with Home and Market Production

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Average-Cost Pricing, Increasing Returns, and Optimal Output in a Model with Home and Market Production Average-cost pricing, increasing returns, and optimal output in a model with home and market production Yew-Kwang Ng Dept. of Economics, Monash University Email: [email protected] Dingsheng Zhang Dept. of Economics, Monash University Institute for Advanced Economic Studies, Wuhan University [email protected] Abstract The analysis of economies of specialization at the individual level by Yang & Shi (1992) and Yang & Ng (1993) is combined with the Dixit & Stiglitz (1977) analysis of monopolistic-competitive firms to show that, ignoring administrative costs and indirect effects (such as rent-seeking), even if both the home and the market sectors are produced under conditions of increasing returns and there are no pre- existing taxes, it is still efficient to tax the home sector to finance a subsidy on the market sector to offset the under-production of the latter due to the failure of price- taking consumers to take account of the effects of higher consumption in reducing the average costs and hence prices, through increasing returns or the publicness nature of fixed costs. Within market production, it is efficient to subsidise more the sector with a higher fixed cost, a lower elasticity of substitution between goods, and a lower degree of importance in preference which all increases the degree of increasing returns. Keywords: increasing returns, average-cost pricing, monopolistic competition, home production, optimal output, fixed costs. JEL Classifications: D43, H21, L11. 1 Average-cost pricing, increasing returns, and optimal output in a model with home and market production The issue of increasing returns is one of those that will be raised incessantly as a neat general solution is lacking and many different outcomes are possible. Increasing returns are also prevalent in the real economy. Arrow (1995) explains how the relevance of information and knowledge in production makes increasing returns prevalent. (See also Wilson 1975, Radner & Stiglitz 1984, Arthur 1994. On empirical evidence for increasing returns, see Ades & Glaeser 1999, Antweiler & Trefler 2002.) Recent interest in the topic has also been considerable, as witnessed by a number of collected volumes (Arrow, et al. 1998, Buchanan & Yoon 1994a, Heal 1999.)1 A model much used in the analysis is one developed by Dixit & Stiglitz (1977) with symmetrical monopolistic competitive firms with free entry and produced with a fixed cost and a constant marginal cost. Though a special case, it reflects much of reality as much of increasing returns may be traced to some big fixed-cost components (a piece of land for farming, a factory for manufacturing production, a shop for retailing, learning costs for many skilled activities, etc.) and the variable costs (raw materials, intermediate goods purchased, stocks ordered, etc.) are largely constant within a wide range. While this model captures much of increasing returns at the firm level, those at the economy level arising from the economies of specialization made possible by the division of labor are analysed by Yang & Ng (1993). This latter framework starts from the most basic level of individual decisions on what activities (home production, trade, employment, consumption, etc.) to undertake to maximize utility and the interaction of the activities of individuals and their implications on economic organization, trade, growth, etc. Though the emergence of firms and other issues (including the choice of the variety of products; see Yang & Shi 1992) are also analysed, the economies of specialization is taken to be confined to the individual level. In principle, one may use this framework to 1 For a survey of the earlier literature on the economics of product variety which is related to monopolistic competition and increasing returns, see Lancaster (1990). 2 model the increasing returns at the firm level through the complicated interaction between different individuals within the firm and their interaction with other factors employed by the firm, but the complication involved may raise issues of manageability. There are also models that analyse the increasing returns from employing production methods with more intermediate goods (e.g. Ethier 1979, Romer 1986, Buchanan & Yoon 1994b). However, there are advantages in directly allowing for increasing returns at the firm level from the fixed-cost components as in the Dixit-Stiglitz model. Moreover, the majority of production in most advanced economies is undertaken by firms with increasing returns prevailing over the whole relevant range of production, but also with individual home production still taking place. The present paper combines the analysis of economies of specialization at the individual level by Yang & Shi (1992) and Yang & Ng (1993) with the Dixit & Stiglitz (1977) analysis of the market production by monopolistic-competitive firms. (For an earlier model combining home production with market production by firms emphasizing the role of the number of intermediate goods and different stages of production, see Locay 1990. Here, the complications due to intermediate goods and stages in production are ignored.) It is hoped that this combination moves a step closer to the real economy with both home and firm/market production. The model developed in Section 1 below could be used to analyse problems other than those discussed in this paper. It is shown in Section 1 that, even if both the home and the market sectors are produced under conditions of increasing returns and there are no pre-existing taxes (including income taxes) on the market sector, it is still efficient to tax the home sector to finance a subsidy on the market sector to offset the under-production of the latter. The home sector is not under-produced because the increasing returns there are fully taken into account by the individuals/households. In the production by firms for the market, as the output is priced at average cost and each consumer takes the price as given, the effect of higher consumption in reducing the price through increasing returns is not taken into account. Viewed differently, the fixed cost of production possesses the publicness characteristic, causing under-production. However, the 3 taxation of the home sector may not be practically feasible. Section 2 allows market production to have two sectors and shows that it is efficient to tax the sector with a lower fixed cost, a higher elasticity of substitution between goods, and a higher degree of importance in preference (as all these factors contribute to a lower degree of increasing returns) and subsidize the other sector. Qualifications on the applicability of these results in the real economy are discussed in the concluding section. 1. A model with home and market production 1.1 The model Consider an economy with M identical consumers. Each of them has the following decision problem for consumption, working, and home production. α β 1−α−β ρ1 ρ1 ρ2 ρ2 (1) Max: u = l [∑ xr ] [∑ x j ] (utility function) r∈R j∈J s.t. ∑ pr xr = w(1−l − ∑l j ) (budget constraint) r∈R j∈J l − a x = j (home production function) j c where pr is the price of good r which are market goods, w is the price of labour, xr is the amount of good r that is purchased from the market, R is the set of market goods, xj is the amount of good j which is home good, lj is the amount of labour used in producing home good j, a<1 is the fixed cost of producing a home good, c is the marginal cost in home production, J is the set of home goods, l is leisure, (1− l − ∑l j ) is the amount of labour hired by firms, ρi ∈(0,1) (open interval) is the j∈J parameter of elasticity of substitution between each pair of consumption goods, α , β are preference parameters, and u is the utility level. It is assumed that each consumer is endowed with one unit of labour, which is the numeraire, so w =1. Each consumer is a price taker and her decision variables are l, lj and xr. It is assumed that the elasticity of substitution 1/(1-ρi) > 1RU! !i >0 for both i = 1, 2. 4 Denoting the number of home goods as m, by symmetry, the budget constraint can be rewritten as follow: (2) ∑ pr xr + mlh + l =1 r∈R where lh is labor used in the production of each home good. )URPWKH/DJUDQJHDQIXQFWLRQ/ ZKHUH LVXVHGDVWKHPXOWLSOLHU RIWKHDERYH maximization problem, the first-order conditions2 are: α −1 β ∂L 1−α−β ρ1 ρ1 ρ1−1 ρ2 lh − a β (3) = 0 ⇒ αl [∑ xr ] xr m ( ) = λ pr ∂xr r∈R c α β −1 ∂L β 1−α−β ρ1 ρ1 ρ2 lh − a β (4) = 0 ⇒ l [∑ xr ] m ( ) = λlh ∂m ρ2 r∈R c α β ∂L 1−α−β ρ1 ρ1 ρ2 lh − a β −1 (5) = 0 ⇒ βl [∑ xr ] m ( ) = cλm ∂lh r∈R c α β ∂L −α−β ρ1 ρ1 ρ2 lh − a β (6) = 0 ⇒ (1−α − β )l [∑ xr ] m ( ) = λ ∂l r∈R c ∂L (2) = 0 ⇒ ∑ pr xr + mlh + l =1 ∂λ r∈R Using (2)-(6), we can get following solutions a (7) lh = 1− ρ2 ρ (1−α − β) (8) l = 2 ρ2 + β(1− ρ2) β (1− ρ ) (9) m = 2 a[ρ2 + β (1− ρ2 )] αρ2 (10) x = ρ r 1 n 1 1−ρ ρ −1 1 1 [ρ2 + β(1− ρ2)]pr (∑ ps ) s=1 where n is the number of market goods. 2 It may be checked that second-order conditions are satisfied. 5 Before we consider the behaviour of firms, we first get the own price elasticity of demand for good r, using (10), we have ∂ln x ρ − n (11) r = 1 ∂ln pr n(1− ρ1) This formula is called Yang-Heijdra formula ( Yang and heijdra, 1993).
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